Why does Wolfram Alpha say that $n/0$ is complex infinity?

Whether or not things are "undefined" largely depeneds on what framework you are working in.

If we are working in the naturals, we might say that $3-5$ is undefined.

There are many systems where it makes sense to assign $\frac{n}{0}$ some value. In this particular example, it is defined to be complex infinity, which can be thought of as follows: suppose we are looking at the complex plane. Similarly to how the complex number "0" is represented by a zero vector of arbitrary direction, we wish to associate all complex numbers of infinite absolute value (regardless of direction) to a single point.

This is complex infinity, and geometrically, by associating all complex numbers of infinite absolute value to be the same on the plane, we have formed a sphere, one with zero on the bottom, and complex infinity on the top. This is called the Riemann Sphere.

Sometimes it is useful in complex analysis to consider the complex numbers plus the "point at infinity". See this wiki article for details: Riemann Sphere